Posteriors

Posteriors, in probability and statistics, refer to the posterior distributions of unknown quantities after observing data. In Bayesian inference, the primary object is the posterior distribution p(theta | D), where theta denotes the parameters of interest and D denotes the observed data. The posterior combines prior beliefs with the information provided by the data.

Bayes' theorem states that p(theta | D) = p(D | theta) p(theta) / p(D), where p(D | theta) is the likelihood, p(theta) is the prior, and p(D) = ∫ p(D | theta) p(theta) dtheta is the marginal likelihood or model evidence. The posterior thus represents updated beliefs about theta after incorporating the data.

Posteriors can sometimes be computed in closed form when the prior and likelihood are conjugate. In most realistic models, numerical methods are required, including Markov chain Monte Carlo, importance sampling, or variational inference. Once the posterior for theta is obtained, one can derive the posterior predictive distribution for new data by integrating over theta: p(y* | D) = ∫ p(y* | theta) p(theta | D) dtheta.

Posteriors are the foundation for uncertainty quantification, producing credible intervals, hypothesis assessments, and decisions under uncertainty. They also enable model comparison via marginal likelihoods or Bayes factors. In hierarchical or multilevel models, posteriors may be inferred for many parameters across levels; in Bayesian nonparametrics, posteriors can be distributions over functions or partitions.

The term "posteriors" may refer to multiple posterior distributions—across parameters, across models, or across different data sets or prior specifications. Limitations include sensitivity to prior choices, model misspecification, and computational demands, especially in high dimensions or complex likelihoods.